For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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2
votes
0answers
303 views

Derivative/Jacobian of the matrix logarithm

I need help finding the Jacobian of the matrix logarithm function, i.e. $\log{M} = R$ defined by $e^R = M = V\begin{bmatrix}e^{\lambda_1} & & \\ & \ddots & \\ & & e^{\lambda_n} ...
3
votes
1answer
60 views

Matrix with rank $1$

Let $A=(a_{ij})_n$ a symmetric matrix with positive coefficients. We suppose that there is $\alpha>0$ such that, for all permutation $\sigma$ of $\{1,\ldots,n\}$, we have ...
2
votes
4answers
10k views

Find the standard matrix for a linear transformation.

If T: $\Bbb R$3→ $\Bbb R$3 is a linear transformation such that $$ T \Bigg (\begin{bmatrix}-2 \\ 3 \\ -4 \\ \end{bmatrix} \Bigg) = \begin{bmatrix} 5\\ 3 \\ 14 \\ \end{bmatrix}, T \Bigg ...
3
votes
5answers
750 views

What does positive definite matrix mean?

What do we mean by a matrix is positive or negative definite? Does it have any analogy with a positive real number?
0
votes
1answer
356 views

Proving that matrix is positive definite

I have a set of real variables $\{a_1,a_2,a_3,...,a_n\}$. Let $u=\sqrt{a_1^2+a_2^2+...+a_n^2}$. I am trying to prove that the following matrix is positive definite: $A\in \mathbb{R}^{n\times n}$, ...
2
votes
2answers
59 views

Are transformation matrices invariant over row operations?

The title says it all. I'm currently taking an introductory course in linear algebra and this issue has not been adressed spesifically. What I'm wondering is this: Given a transformation matrix $A$ ...
2
votes
1answer
124 views

Derivative of the off-diagonal $L_1$ matrix norm

We define the off-diagonal $L_1$ norm of a matrix as follows: for any $A\in \mathcal{M}_{n,n}$, $$\|A\|_1^{\text{off}} = \sum_{i\ne j}|a_{ij}|.$$ So what is $$\frac{\partial ...
2
votes
0answers
55 views

positive definite matrix and double non-negative matrix with 0-1 entries

If we have a positive definite (strictly) matrix $A$ and $M$ semi-positive definite with entries $0$ or $1$ and diagonal all ones, What are the conditions to have $\operatorname{max eigenvalue}(A ...
3
votes
2answers
415 views

How to compute a matrix for rotating and centering rectangle in viewport?

I have a rectangle given by 4 points. I'm trying to compute a transformation matrix such that the rectangle will appear straight and centered within my viewport. I'm not even sure where to begin. ...
2
votes
2answers
396 views

square root of a real matrix

I want to compute the square root of a real symmetric positive definite matrix $S\in \mathcal{M}_{m,m}$ such that $S^{1/2}S^{1/2}=S$ and it's well known that this decomposition is unique. My question ...
1
vote
1answer
88 views

Counting degrees of freedom

Why does a tensor with 3 indices in $n$ dimensions, such that if you swap any two of the indices gives the same value, has degree of freedom equal to $$n+2 \choose 3$$? I would have thought that it's ...
3
votes
1answer
56 views

How to construct non-square isometry matrix

How can we construct a non-square isometry matrix $U\in \mathcal{M_{n,m}}$; that is, all columns of $U$ are orthonormal and $U U^T=I_{n,n}$?
1
vote
1answer
570 views

Find 3D rotation vector and angle to transform a rectangle into a given quadrilateral

I have a given rectangle that I need to transform into a given quadrilateral shape that resulted from a rotation and translation in 3D space, and subsequent projection. ...
2
votes
2answers
84 views

Rank of a bilinear form

I have to prove that a bilinear form $B$ has full rank, and I would like to know some ideas on how to prove that. Can anyone give an idea?
2
votes
1answer
256 views

Matrix norms involving singular values

The induced 2-norm of a $m \times n$ matrix $A$ is known as $$ ||A||_2 = \underset{||x||=1}{\text{max}} ||Ax||_2 = \sigma_1,$$ where $\sigma_1$ is the largest singular value of $A$. Then, is it ...
0
votes
0answers
332 views

Prove that if its reduced row echelon form is [R c] then R is the reduced row echelon form of A.

Let $[A\;b]$ be the augmented matrix of a system linear equations. Prove that if its reduced row echelon form is $[R\;c]$, then $R$ is the reduced row echelon form of $A$. How do I prove it? I mean ...
0
votes
1answer
44 views

Differential and derivative of $X^{-2}$

Determine the differential and derivative of $F(X) = X^{-2}$ in which the variable X is an n x n-matrix. I computed the differential by using the product rule. So I first wrote $$ f(X)= X^{-1} X^{-1} ...
-1
votes
1answer
128 views

Elementary lower-triangular $4\times 4$ matrices

What are the three elementary lower triangular $4 \times 4$ matrices and what does their operation do? How can I prove that for all of these, $\det(L)=1$ and $L(x)^{-1}=L(-x)$?
4
votes
2answers
71 views

A quantitative measure of rank of a matrix

The rank of a matrix is only defined as integers. Is there some other criteria that is more quantitative. E.g. $$A = \begin{bmatrix} 1 & 1\\ 1 & 0\\ \end{bmatrix} $$ $$B= \begin{bmatrix} 1 ...
8
votes
1answer
305 views

How to diagonalize this matrix?

Consider the $n\times m$ matrix $M=[M_1, \ldots, M_m]$ where the $i$-th column reads $$ M_i= \,^t(\underbrace{1,\ldots,1}_{a_i},0,\ldots,0) $$ where the $a_i$'s are given positive natural numbers. ...
4
votes
3answers
75 views

How to show $T$ is not one-one and $T$ is not ont0?

Suppose $V$ is the space of all $n \times n$ matrices with real elements. Define $T : V \to V$ by $$T (A) = AB − BA,\; A \in V,$$ where $B \in V$ is a fixed matrix. Show that for any $B \in V$, ...
1
vote
2answers
583 views

Differential and derivative of the trace of a matrix

If $X$ is a square matrix, obtain the differential and the derivative of the functions: $f(X) = \operatorname{tr}(X)$, $f(X) = \operatorname{tr}(X^2)$, $f(X) = \operatorname{tr}(X^p)$ ...
0
votes
2answers
70 views

Any idea which matrix theorem this is?

I came across a theorem that boyd uses to convert the simplex to the form of a polyhedra. I don't know anything about this theorem. Theorem states: If $B$ has rank $k$, then we can find two matrices ...
2
votes
3answers
46 views

Matrix decomposition again

If some matrix (M×N) can be expressed as product of (M×1) and (1×N) vectors: what is proper term for such kind of decomposition? how to tell if such kind of decomposition exists for given matrix? ...
3
votes
0answers
72 views

Elementary Lower Matrices

First of all forgive me for my lack of format. I want to prove that the following elementary lower triangular $nxn$ matrix $Li(x)= I-xe(i)^T$ where $x=[0 \ldots 0 x(i+1) \ldots x(n)]^T$ has the ...
-2
votes
1answer
280 views

Prove the following facts about the matrix exponential:

a) Prove that $(e^{At}-I)/t\rightarrow A$ as $t\rightarrow 0$, meaning $||(e^{At}-I)/t - A|| \rightarrow 0$ as $t\rightarrow 0$ for all $A\in\mathbb{C^{n \times n}}$. Hint: You may use the inequality ...
5
votes
3answers
644 views

Differentiate $f(x)=x^TAx$

Calculate the differential of the function $f:\Bbb R^n\rightarrow\Bbb R$ given by $f(x)=x^TAx$, with $A$ symmetric. Also differentiate this function to $x^T$. How exactly does this work in the case ...
0
votes
1answer
77 views

Lower bound on the norm of product of non square matrices

The following inequality is known: $\parallel AB\parallel\geq\parallel A\parallel \sigma_{n}(B)$. However, it is only valid where both $A$ and $B$ are square. Is there an analogue for rectangular ...
0
votes
3answers
151 views

How is a $4\times 4$ Matrix built (concerning position, translation, scale and rotation)?

I am given to understand that a $4\times 4$ matrix can contain position, translation, scale and rotation, but I don't know where all of these are in the matrix. What I have seen so far is that ...
-1
votes
2answers
105 views

three questions on analytic geometry and matrices

the lines $x-2y=4$ and $6x+ay=8$ are perpendicular. Calculate the value of $a$. prove that the matrix $\pmatrix{\cos\theta& \sin\theta\\ -\sin\theta & \cos\theta}$ ...
1
vote
1answer
46 views

Conjugacy class in matrix ring

Let $M_{2}(\mathbb{R})$ be the ring of $2\times 2$ matrices over the reals and $M_{2}(\mathbb{R})^*$ the set of invertible such matrices. Consider any $A \in M_{2}(\mathbb{R})$ such that $ A^{2}=-I$, ...
2
votes
1answer
76 views

Is there a matrix satisfying a certain condition

I have a numerical problem which boils down to the following: We are given a square matrix $R$, with a bunch of zeros in it. We want to check if there exists orthonormal matrix $T$ such that $TT'=I$, ...
9
votes
1answer
265 views

What can we say about $(I-AD)^{-1}$ if $D$ is a diagonal matrix?

Assume we know that square matrices $A$ and $(I-AD)^{-1}$ are invertible and also $D$ is a diagonal matrix. Also assume that $A$ is a symmetric matrix. My question is when we can express ...
3
votes
3answers
146 views

Fit a quadratic form given covariant derivatives on the sphere?

I am trying to solve for a particular vector given covariant first and second derivative for a function on a sphere. If you have a quadratic form restricted to the sphere: $f(x) = ...
0
votes
1answer
229 views

How to generate an observation matrix from its covariance matrix

I would like to know how to generate a matrix $A$ from its covariance matrix whatever the property of this matrix, i.e. correlated or uncorrelated. Specifically I'm interested to generate a noise ...
0
votes
3answers
126 views

Simple linear algebra question

Let's say $A$ is an orthogonal $2\times2$ matrix over $\bf C$ and not diagonalizable over $\bf C$. Why then the determinant of $A$ must be $1$? I guess I'm missing something easy...
1
vote
2answers
54 views

assume $A\in M_n(\mathbb C)$,$A^2=0$ how prove $\exists C,B\in M_n(\mathbb C)$ such that A=BC and CB=0?

assume $A\in M_n(\mathbb C)$,$A^2=0$ how prove $\exists C,B\in M_n(\mathbb C)$ such that $A=BC$ and $CB=0$. Thanks in advance
1
vote
2answers
41 views

How to find $a$ and $b$

I require to find $a$ and $b$ in the following: $$\begin{bmatrix}0\\-40\\25\end{bmatrix}=a\begin{bmatrix}1\\-4\\3\end{bmatrix}+b\begin{bmatrix}1\\4\\-2\end{bmatrix}$$ Now I moved the $b[\ ]$ group to ...
4
votes
3answers
730 views

Bounds on off-diagonal entries of a correlation matrix

Assume that all the entries of an $n \times n$ correlation matrix which are not on the main diagonal are equal to $q$. Find upper and lower bounds on the possible values of $q$. I know that the ...
4
votes
2answers
203 views

An example of a $4×4$ matrix $A$ such that $A\not= I$, $A^2\not=I$, …, $A^5 = I$

How do I go about solving this? I went for tutoring and the tutor said I am trying to get to an Identity matrix so I should start with an identity matrix and mix the values around till I get a ...
5
votes
1answer
163 views

Centralizers of Matrices

let $A$ be a complex matrix. Denote by $J(A)$ the Jordan Canonical Form of $A$. Let $C[J(A)]$ be the centralizer of $J(A)$ in $M_n(\mathbb C)$. Can we construct a real matrix $B$, that is, $B$ has ...
2
votes
3answers
174 views

Proving that a set is linearly independent when vector is not in the Span.

I understand how to do this when I have values for the vectors, but what if there are no values? I also know that if the solution is trivial, it is independent. Basically, can I solve this with Gauss ...
2
votes
1answer
881 views

Inverse of orthogonal projection

I have an $N \times N$ orthogonal projection matrix $P = A^H(AA^H)^{-1}A$ that I'm trying to find the inverse for. I'm using matlab, however, I keep getting the warning "the matrix is close to ...
2
votes
1answer
636 views

The rank of a block matrix as a function of the rank of its submatrices.

I would like to post this problem here in this forum. Having the following block matrix: \begin{equation} M=\begin{bmatrix} S_1 &C\\ C^T &S_2\\ \end{bmatrix} \end{equation} I would like to ...
1
vote
1answer
456 views

Derivative of matrix inverse

I am trying to find the derivative of a matrix with respect to the inverse of the same matrix. The matrix in question is a non singular symmetric matrix. Any thoughts?
1
vote
0answers
30 views

Sum of spatio-temporal window

I would like to get confirmation about a formula. It looks quite simple but I can't make it work, so I start to have some doubt. Let's say I have 2 images $I_t$ and $I_{t-1}$ which are 2 consecutives ...
0
votes
1answer
52 views

Transpose of 2 matrices together

So if I have an $m\times n$ matrix $A$ and I represent that matrix as $\displaystyle A = QR$, how do I write $A^{T}$ (transpose) in terms of the original $\displaystyle QR$? Does it become ...
3
votes
1answer
229 views

An inequality involving the norms of symmetric positive definite matrices

Given A and B two real symmetric positive definite matrices is it true that, for some norm $\|.\|$, this inequality holds $$ \|AB-I\| \leq \|A^2B^2-I\| \qquad ? $$
3
votes
1answer
520 views

Product of positive-definite matrices has positive trace

In his paper "On the existence of a connection with curvature zero", Milnor makes the following claim. Let $X$ and $Y$ be positive-definite $n \times n$ matrices. Then the trace of $X Y$ is ...
2
votes
2answers
99 views

Is my solution to the system of equations correct?

If I'm told that $T(\vec x)=A\vec x=\vec b$ and $A=\left[\begin{matrix}1&-3&2\\ 3&-8&8\\ 0&1&2\\ 1&0&8\\\end{matrix}\right]$ and that $\vec ...